|
%I #24 Jul 31 2022 07:50:56
%S 1,4,11,23,41,66,99,141,193,256,331,419,521,638,771,921,1089,1276,
%T 1483,1711,1961,2234,2531,2853,3201,3576,3979,4411,4873,5366,5891,
%U 6449,7041,7668,8331,9031,9769,10546
%N Expansion of (1+x^2-x^3)/(1-x)^4.
%C If Y is a 3-subset of an n-set X then, for n>=4, a(n-4) is the number of (n-3)-subsets of X which do not have exactly one element in common with Y. - _Milan Janjic_, Dec 28 2007
%H G. C. Greubel, <a href="/A027378/b027378.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = binomial(n+4, 3) - 3*(n+1). - _Milan Janjic_, Dec 28 2007 [Correction by _Mathew Englander_, Feb 03 2022]
%F a(n) = A006503(n) + 1 = A034857(n) + 5 = A116721(n+2) - 1 = A006416(n+1) + 3. - _Mathew Englander_, Feb 03 2022
%F E.g.f.: (1/6)*(6 + 18*x + 12*x^2 + x^3)*exp(x). - _G. C. Greubel_, Jul 30 2022
%t CoefficientList[Series[(1+x^2-x^3)/(1-x)^4,{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{1,4,11,23},50] (* _Harvey P. Dale_, May 17 2021 *)
%o (Magma) [(n^3 +9*n^2 +8*n +6)/6: n in [0..50]]; // _G. C. Greubel_, Jul 30 2022
%o (SageMath) [(n^3 +9*n^2 +8*n +6)/6 for n in (0..50)] # _G. C. Greubel_, Jul 30 2022
%Y Cf. A006416, A006503, A034857, A116721.
%Y Appears to be first differences of A252814.
%Y First differences at A027379 (omitting first term).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
|
